The Power of Arc Consistency for CSPs Defined by Partially-Ordered Forbidden Patterns

Abstract

Characterising tractable fragments of the constraint satisfaction problem (CSP) is an important challenge in theoretical computer science and artificial intelligence. Forbidding patterns (generic sub-instances) provides a means of defining CSP fragments which are neither exclusively language-based nor exclusively structure-based. It is known that the class of binary CSP instances in which the broken-triangle pattern (BTP) does not occur, a class which includes all tree-structured instances, are decided by arc consistency (AC), a ubiquitous reduction operation in constraint solvers. We provide a characterisation of simple partially-ordered forbidden patterns which have this AC-solvability property. It turns out that BTP is just one of five such AC-solvable patterns. The four other patterns allow us to exhibit new tractable classes. CC Creative Commons COOPER ANDŽIVNÝA substantial body of work exists from the past two decades on applications of universal algebra in the computational complexity of and the applicability of algorithmic paradigms to CSPs. Moreover, a number of celebrated results have been obtained through this method; see [2] for a recent survey. However, the algebraic approach to CSPs is only applicable to language-based CSPs, that is, classes of CSPs defined by the set of allowed constraint relations but with arbitrary interactions of the constraint scopes. For instance, the well-known 2-SAT problem is a class of language-based CSPs on the Boolean domain {0, 1} with all constraint relations being binary, that is, of arity at most two.On the other side of the spectrum are structure-based CSPs, that is, classes of CSPs defined by the allowed interactions of the constraint scopes but with arbitrary constraint relations. Here the methods that have been successfully used to establish complete complexity classifications come from graph theory [28,33].The complexity of CSPs that are neither language-based nor structure-based, and thus are often called hybrid CSPs, is much less understood; see [9,20] for recent surveys. One approach to hybrid CSPs that has been rather successful studies the classes of CSPs defined by forbidden patterns; that is, by forbidding certain generic subinstances. The focus of this paper is on such CSPs. We remark that we deal with binary CSPs but, unlike in most papers on (the algebraic approach to) language-based CSPs, the domain is not fixed and is part of the input.An example of a pattern is given in Figure 1(a) on page 5. This is the so-called broken triangle pattern (BTP) [16] (a formal definition is given in Section 2). BTP is an example of a tractable pattern, which means that the class of all binary CSP instances in which BTP does not occur is solvable in polynomial time. The class of CSP instances defined by forbidding BTP includes, for instance, all tree-structured binary CSPs [16]. There are several generalisations of BTP, for instance, to quantified CSPs [26], to existential patterns [10], to patterns on non-binary constraints [18], and other classes [34,17].The framewor...

Extending Broken Triangles and Enhanced Value-Merging

“…However, by imposing some restrictions on the constraint scopes and/or relations, we can define tractable classes of instances which can be solved in polynomial time. The BTP (Broken Triangle Property) tractable class, is an important tractable class since it generalises certain previously known classes based exclusively on properties of the constraint scopes or the constraint relations and has been the inspiration for a new branch of research on tractable classes of CSPs based on forbidden patterns [1,4,9,11,18]. The Broken Triangle Property imposes the absence of so-called broken triangles.…”

The power of propagation: when GAC is enough

“…CSP instances satisfying the BTP are characterised by the absence of a certain kind of pattern of allowed and disallowed combinations on three variables [19]. Four additional patterns of this kind whose absence guarantees that a binary CSP instance is decided by GAC have been identified in [19].…”

“…Four additional patterns of this kind whose absence guarantees that a binary CSP instance is decided by GAC have been identified in [19].…”

On Singleton Arc Consistency for CSPs Defined by Monotone Patterns

“…Of key importance amongst these pre-processing algorithms are the relatives of arc consistency propagation including generalised arc consistency (GAC) and singleton arc consistency (SAC). Surprisingly there are large classes [16,23,13,28] of the CSP for which GAC or SAC are decision procedures: after establishing consistency if every variable still has a non-empty domain then the instance has a solution.More generally, these results fit into the wider area of research aiming to identify subproblems of the CSP for which certain polynomial-time algorithms are decision procedures. Perhaps the most natural ways to restrict the CSP is to limit the constraint relations that we allow or to limit the structure of (the hypergraph of) interactions of the constraint scopes.…”

“…Of key importance amongst these pre-processing algorithms are the relatives of arc consistency propagation including generalised arc consistency (GAC) and singleton arc consistency (SAC). Surprisingly there are large classes [16,23,13,28] of the CSP for which GAC or SAC are decision procedures: after establishing consistency if every variable still has a non-empty domain then the instance has a solution.…”